integral calculus 271

information theory The branch of

PROBABILITY

the-

ory that deals with the transmission, processing, and

checking of messages sent electronically is called infor-

mation theory. The field was established in 1948 by

American mathematician C

LAUDE

E

DWARD

S

HANNON

,

who first showed that it is possible to encode all types of

information (words, sounds, pictures) into simple

sequences of 0 and 1 bits that can then be transmitted

along a wire as pulses. (Up to then, scientists thought it

would be necessary to transmit electromagnetic waves

along wires.) To adjust for erroneous noise that is often

transmitted along with the message, mathematicians

have since developed probability techniques that deter-

mine the likelihood that the message received is free

from errors. Over the decades, mathematicians have also

developed efficient redundancy checks to help detect and

correct errors. These techniques are far more efficient

than simply repeating the message for comparison.

Information theory is also used to measure the

amount of information a signal of any kind might

contain. Transmitting and correctly receiving a single

letter of the alphabet—that is, one of any 26 letters—

contains more information, for instance, than the

receipt of a single binary digit, 0 or 1. Mathemati-

cians use logarithms to measure information content

and say that receipt of a letter of the alphabet contains

as much information as receipt of a

binary digit. (We have .) This assumes that

each letter in the alphabet is equally likely to occur. In

general, if the probability of the letter abeing sent is

p1, the letter bis p2, and so forth, then the measure of

informational content is given by:

– p1log2p1– p2log2p2–…– p26 log2p26

(This agrees with the previous value of 4.7 computed

when each probability pihas value 1/26.) This quantity

is a measure of the entropy of the initial data set.

The field of information theory has obvious applica-

tions to work in telegraphy, radio transmission, and the

like, but it has also recently been used to analyze human

speech, in the study of languages, and in cybernetics.

instantaneous value The value of a varying quan-

tity, such as

VELOCITY

, at a particular instant in time is

called its instantaneous value. An instantaneous value

is a

DERIVATIVE

.

integer (directed number, signed number) Any of the

positive or negative

WHOLE NUMBER

s, or

ZERO

, is called

an integer: …–3,–2,–1,0,1,2,3,… More precisely, once

the

NATURAL NUMBER

s have been defined via P

EANO

’

S

POSTULATES

, say, one can define an integer to be any

quantity that can be expressed as the sum or difference

of two natural numbers. (For instance, the number –5

can be regarded, formally, as the pair of natural num-

bers 3 and 8 written in the form 3–8. It can be defined

equally well by the pair 1–6 or the pair 12–17, for

example.) The difference of any two integers is always

another integer, which constitutes a mathematical

RING

.

The set of integers is denoted Z(from the German word

Zahlen for “numbers”). German mathematician G

EORG

C

ANTOR

(1845–1918) showed that the set of integers is

COUNTABLE

and so has

CARDINALITY

ℵ0.

The T

AYLOR SERIES

of has the

integers appearing as coefficients:

(This series is valid for –1 < x< 1.) This shows, for

instance, that in setting x= , the fraction has the

integers appearing in turn in each decimal place:

= 0.12345.... (Unfortunately our practice of “carry-

ing digits” disguises the fact that the pattern we see

continues.)

See also

FLOOR

/

CEILING

/

FRACTIONAL PART FUNC

-

TIONS

.

integral See

ANTIDIFFERENTIATION

;

INTEGRAL CAL

-

CULUS

.

integral calculus The calculation of sums of infi-

nitely small quantities,

INFINITESIMAL

s, is called inte-

gral calculus. For example, consider the problem of

finding the length of a curved path using only a straight

yardstick. One could approximate the distance along

the curve by marking a number of points along the

curve, measuring the straight-line segments between

them, and summing the lengths of these segments.

10

–

81

10

–

81

1

–

100